# Help -- Quadratic Equations and Fractions

#### lovezoey101

##### New member
how do you solve these?
havent learned this but it was assigned for homework and i cant find any examples like these in my book.
thank you.

What two numbers are represented by each of the following?
4 plus minus symbol 1
4/3 plus minus symbol i square root 2/3
11 plus minus symbol square root of 2

Give the first step in the solution of each equation.
a. x squared = 5 b. (x-3) squared = 5 c. (x+3) squared = 5 d. (2x+3) squared = 5

What must be added to make the given expression a trinomial square?
x squared+ 2x+?
y squared + 2/5y+?
t squared -t+?

#### fasteddie65

##### Full Member
"What two numbers are represented by each of the following?"

4 plus minus symbol 1 = (I think) 4 + (-1) = 3
4/3 plus minus symbol i square root 2/3 = 4/3 + ?(2/3) i
11 plus minus symbol square root of 2 = 11 ± ?2

Give the first step in the solution of each equation.
a. x squared = 5
Take the square root of both sides.

b. (x-3) squared = 5
Take the square root of both sides.

c. (x+3) squared = 5
Take the square root of both sides.

d. (2x+3) squared = 5
Take the square root of both sides.

What must be added to make the given expression a trinomial square?
x squared+ 2x+? 1
y squared + 2/5y+? 1/25 = (1/5)^2
t squared -t+? (1/4) = (-1/2)^20

#### fasteddie65

##### Full Member
The last should be (1/2)^2.

#### lovezoey101

##### New member
thanks
could you explain how you figure out which two numbers are represented in the first part?
and the second part a-d are you sure that for all the problems its square both sides?
and how did you do the last few?

#### mmm4444bot

##### Super Moderator
Staff member
lovezoey101 said:
... What two numbers are represented by each of the following?

4 plus minus symbol 1

I'm not sure why Fast Eddie guessed that the phrase "plus minus symbol" means different things for different exercises, but I'll go with his other guess.

4 +/- 1

Okay, Zoey, if this guess regarding your notation is correct, then please tell me why you cannot complete the following arithmetic.

4 + 1 = ?

4 - 1 = ?

#### lovezoey101

##### New member
So it would be 5 and 3?
but what do you do if there is a square root?
im just really confused ive never had it explained to me.

and yes plus minus sign is ±

#### mmm4444bot

##### Super Moderator
Staff member
lovezoey101 said:
So it would be 5 and 3? ... For the first exercise, yes.

... ive never had it explained to me ... That's not fair.

The ± sign allows us to write two expressions at once; you need to separate the + and - signs into two separate operations to calculate both values.

If the two quantities being added (or subtracted) cannot be combined, then the expression is already simplified.

EG:

4 - ?3 is one value, and

4 + ?3 is another.

Neither of these two values can be simplified; we cannot do the arithmetic.

They can both be written as 4 ± ?3.

Of course, we are able combine multiples of roots.

EG:

4?3 + ?3 = 5?3

4?3 - ?3 = 3?3

#### lovezoey101

##### New member
oh thank you sooo much
it makes complete sense

#### mmm4444bot

##### Super Moderator
Staff member
lovezoey101 said:
... and the second part a-d are you sure that for all the problems its square both sides? ...

I think that you need to read Fast Eddie's answers again more carefully. He did not write, "square both sides".

Since the symbol x appears inside parentheses, we need to get rid of the parentheses before we can isolate x. Therefore, the first step is to get rid of the parentheses by taking the square root of both sides.

EG:

$$\displaystyle (x + 4)^2 = 100$$

Take the square root of both sides.

$$\displaystyle |x + 4| = \sqrt{100}$$

Consider both the principle square root of 100, as well as its opposite, to get rid of the absolute value symbols.

$$\displaystyle x + 4 = \pm \sqrt{100}$$

$$\displaystyle x + 4 = \pm10$$

Even though parts (a) through (d) do not require you to solve for x, I'll continue to do so because you just learned the ± symbol; I would like you to see it used again.

To solve for x, we need to get rid of the 4 on the left side of the equation. So we subtract it. We subtract it from both sides of the equation.

$$\displaystyle x = \pm10 - 4$$

$$\displaystyle x = 10 - 4$$

$$\displaystyle x = -10 - 4$$

So, the two solutions for (x + 4)[sup:i9vcq7wi]2[/sup:i9vcq7wi] = 100 are x = 6 and x = -14.

#### mmm4444bot

##### Super Moderator
Staff member
lovezoey101 said:
... and how did you do the last few?

(Do a Google search on keywords "completing the square", if you would like to see lessons.)

Take one half of the first-degree term's coefficient, and square the result.

EG:

x[sup:1sy40ho3]2[/sup:1sy40ho3] + bx

(b/2)[sup:1sy40ho3]2[/sup:1sy40ho3] = b[sup:1sy40ho3]2[/sup:1sy40ho3]/4

b[sup:1sy40ho3]2[/sup:1sy40ho3]/4 is the term that we need to add to the expression x[sup:1sy40ho3]2[/sup:1sy40ho3] + bx in order to get a perfect square trinomial.

x^2 + bx + b[sup:1sy40ho3]2[/sup:1sy40ho3]/4 is a perfect square trinomial because it's the square of (x + b/2).

Use FOIL to expand (x + b/2)[sup:1sy40ho3]2[/sup:1sy40ho3], and you'll get x^2 + bx + b[sup:1sy40ho3]2[/sup:1sy40ho3]/4.